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| In [[coding theory]], the '''Singleton bound''', named after Richard Collom Singleton, is a relatively crude bound on the size of a [[block code]] <math>C</math> with block length <math>n</math>, size <math>r</math> and minimum distance <math>d</math>.
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| ==Statement of the Bound==
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| The minimum distance of a set <math>C</math> of codewords of length <math>n</math> is defined as
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| :<math>d = \min_{x,y \in C, x \neq y} d(x,y)</math>
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| where <math>d(x,y)</math> is the [[Hamming distance]] between <math>x</math> and <math>y</math>. The expression <math>A_{q}(n,d)</math> represents the maximum number of possible codewords in a q-ary block code of length <math>n</math> and minimum distance <math>d</math>.
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| Then the Singleton bound states that
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| :<math>A_q(n,d) \leq q^{n-d+1}.</math>
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| ==Proof==
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| First observe that there are <math>q^n</math> many q-ary words of length <math>n</math>, since each letter in such a word may take one of <math>q</math> different values, independently of the remaining letters.
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| Now let <math>C</math> be an arbitrary q-ary block code of minimum distance <math>d</math>. Clearly, all codewords <math>c \in C</math> are distinct. If we delete the first <math>d-1</math> letters of each codeword, then all resulting codewords must still be pairwise different, since all original codewords in <math>C</math> have [[Hamming distance]] at least <math>d</math> from each other. Thus the size of the code remains unchanged.
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| The newly obtained codewords each have length
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| :<math>n-(d-1)=n-d+1</math> | |
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| and thus there can be at most
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| :<math>q^{n-d+1}</math>
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| of them. Hence the original code <math>C</math> shares the same bound on its size <math>|C|</math>:
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| :<math>|C| \le A_q(n,d) \leq q^{n-d+1}.</math>
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| ==MDS codes== | |
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| Block codes that achieve equality in Singleton bound are called '''MDS (maximum distance separable) codes'''. Examples of such codes include codes that have only one codeword (minimum distance n), codes that use the whole of <math>(F_{q})^{n}</math> (minimum distance 1), codes with a single parity symbol (minimum distance 2) and their [[dual code]]s. These are often called ''trivial'' MDS codes.
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| In the case of binary alphabets, only trivial MDS codes exist.<ref>see e.g. Vermani (1996), Proposition 9.2.</ref>
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| Examples of non-trivial MDS codes include [[Reed–Solomon error correction|Reed-Solomon codes]] and their extended versions.<ref> see e.g. MacWilliams and Sloane, Ch. 11.</ref>
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| ==See also==
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| *[[Gilbert–Varshamov bound]]
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| *[[Plotkin bound]]
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| *[[Hamming bound]]
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| *[[Johnson bound]]
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| *[[Griesmer bound]]
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| ==Notes==
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| <references/> | |
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| ==References==
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| * {{cite journal | author=R.C. Singleton | title=Maximum distance q-nary codes | journal=IEEE Trans. Inf. Theory | volume=10 | pages=116–118 | year=1964 | doi=10.1109/TIT.1964.1053661 | issue=2 }}
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| '''Further reading'''
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| * {{cite book | author=J.H. van Lint | authorlink=Jack van Lint | title=Introduction to Coding Theory | edition=2nd | publisher=Springer-Verlag | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | date=1992 | isbn=3-540-54894-7 | page=61 }}
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| * {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=[[Neil Sloane|N.J.A. Sloane]] | title=The Theory of Error-Correcting Codes | publisher=North-Holland | date=1977 | isbn=0-444-85193-3 | pages=33,37 }}
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| * {{cite book | last1=Niederreiter | first1=Harald | last2=Xing | first2=Chaoping | title=Rational points on curves over finite fields. Theory and Applications | series=London Mathematical Society Lecture Note Series | volume=285 | location=[[Cambridge]] | publisher=[[Cambridge University Press]] | year=2001 | chapter=6. Applications to algebraic coding theory | isbn=0-521-66543-4 | zbl=0971.11033 }}
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| * L. R. Vermani: Elements of algebraic coding theory, Chapman & Hall, 1996.
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| {{DEFAULTSORT:Singleton Bound}}
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| [[Category:Coding theory]]
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| [[Category:Inequalities]]
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| [[Category:Articles containing proofs]]
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